EVAT 554 OCEAN-ATMOSPHERE DYNAMICS EQUATIONS OF MOTION (CONT); ENERGY EQUATION LECTURE 4 (Reference: Peixoto & Oort, Chapter 3)

Zonal Momentum Balance: Meridional Momentum Balance: Vertical Momentum Balance: Continuity

SIMPLIFYING APPROXIMATIONS 1. “Boussinesq Approximation” (accept in gravity or “buoyancy” term) 2. Ignore “Metric Terms” (terms that scale as 1/a are orders of magnitude smaller than other terms) 3. Assume equations averaged over e.g. several hours (replace molecular diffusion with Eddy Diffusion based on contribution of averaged non-linear terms)

Zonal Momentum Balance: Meridional Momentum Balance: Vertical Momentum Balance: Continuity: But this is not a closed set of equations!

First Law of Thermodynamics Energy is neither destroyed nor created, but changes form during ordinary physical and chemical processes CONSERVATION OF ENERGY Heating = Change in Internal Energy + Work Done

CONSERVATION OF ENERGY First Law of Thermodynamics Energy is neither destroyed nor created, but changes form during ordinary physical and chemical processes Heating = Change in Internal Energy + Work Done

Includes radiative heating, latent heating, frictional heating, conduction and turbulent heat flux (“diabatic” heating) Heating = Change in Internal Energy + Work Done CONSERVATION OF ENERGY First Law of Thermodynamics Energy is neither destroyed nor created, but changes form during ordinary physical and chemical processes

Heating = Change in Internal Energy + Work Done CONSERVATION OF ENERGY Example: How much energy is needed to warm 2 kg of dry air by 5 o C? m air = 2 kg,  T = 5 o C  Q H = m air C p  T= (2kg)(1005 J kg -1 K -1 )(5 o C) = kJ Cp = 1005 J kg-1 K-1 (dry air)

CONSERVATION OF ENERGY Define the heating rate,

CONSERVATION OF ENERGY Define the heating rate, Combine molecular and eddy diffusive heat transport:

CONSERVATION OF ENERGY Combine molecular and eddy diffusive heat transport:

CONSERVATION OF ENERGY

Boundary Terms

CONSERVATION OF ENERGY Planck Blackbody Incident Solar

We still do not have a closed system of equations! equation of state... Let us first consider the atmosphere…